# Category: Mathematics

Pythagoras Theorem: In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Given: A right-angle triangle ABC in which ∠B = 90° To prove: AC2 = AB2 + BC2 Construction: From B draw BD ⊥ AC Proof: In triangles ADB and ABC, we have∠ADB …

Areas of Two Similar Triangles: Theorem 1: Prove that the ratio of the areas of two similar triangles is equal to the ratio of the square of any two corresponding sides. Given: △ABC ~ △DEFSo, AB/DE = BC/EF = AC/DFAlso, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F To prove: area △ABC/area △DEF = AB2/DE2 …

Criteria For Similarity of Two Triangles: We know that two triangles are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio. That is, if in two triangles ABC and PQR, (i) ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R, and (ii) AB/PQ = BC/QR = CA/RP, …

Basic Proportionality Theorem: Theorem 1: In a triangle a line drawn parallel to one side, to intersect the other two sides then it divides the two sides in the same ratio. Given: A △ABC in which DE || BC and intersects AB and AC in D and E respectively. To prove: AD/DB = AE/EC Construction: Join …

Linear Inequations in Two Variables: A statement of inequality containing two variables is known as a linear inequation of two variables. Examples of linear inequations of two variables are- (i) ax + by + c > 0 (ii) ax + by + c < 0 (iii) ax + by + c ≥ 0 (iv) ax + …

Linear Inequations in One Variable: We know that equations are mathematical statements having two sides connected by a sign of equality. In an inequation, the two sides of the statement are connected by a sign of inequality. The signs of inequality are <, >, ≤, ≥, ≠. Statements like 2x < 3, 12x – 7 ≤ …

Quadratic Equations Common Roots: Let the two quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have a common root α. Then, α will satisfy both the equations ∴ a1α2 + b1α + c1 = 0 and a2α2 + b2α + c2 = 0 By the method of …

Formation of Quadratic Equations With Given Roots: Let α and β be the roots of the equation ax2 + bx + c = 0, where a ≠ 0. Then, α + β = -b/a and αβ = c/a. Now, the equation can be written as- x2 +(b/a)x + c/a = 0 or, x2 – (-b/a)x + c/a = …

Nature of Roots of Quadratic Equation: The expression, b2 – 4ac, connecting the coefficients of the quadratic equation ax2 + bx + c = 0, with a ≠ 0 determines the nature of the roots α and β of the equation and is known as the discriminant of the equation. It is generally denoted by D. Several …

Roots and Coefficients of a Quadratic Equations: The roots of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) are given by, Example 1- If one of the roots of the equation ax2 + bx + c = 0 is three times the other, then prove that 3b2 = 16 ac. Solution- The given …